Question

How is the intensity of emitted radiation defined? For a diffusely emitting surface, how is the emissive power related to the intensity of emitted radiation?

Short answer

Answer: For a diffusely emitting surface, the emissive power (E) is related to the intensity of emitted radiation (I_0) by the equation E = 2πI_0.

Step-by-step solution

Define Intensity of Emitted Radiation

The intensity of emitted radiation (I) is defined as the power per unit solid angle emitted in a specified direction, divided by the area of the source orthogonal to the direction of the radiation. Mathematically, it can be written as: I = \frac{dP}{dA\cdot d\Omega} where dP is the infinitesimal power emitted in the direction of the solid angle element dΩ, and dA is the infinitesimal source area element perpendicular to the radiation direction.

Define Emissive Power

Emissive power (E) is the total power radiated per unit area of a surface, given by the integral of the intensity of radiation over all angles in the hemisphere above the surface: E = \int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}}I(\theta, \phi)\, sin\theta\: d\theta\, d\phi where I(θ, φ) is the intensity of emitted radiation in the direction specified by the polar angle θ and azimuthal angle φ.

Diffuse Emission and its Relation to Intensity

For a diffusely emitting surface, the intensity of emitted radiation is the same in all directions. This means that I(θ, φ) is independent of θ and φ, and thus equals a constant value I_0: I(\theta, \phi) = I_0 To determine the emissive power for a diffusely emitting surface, substitute the constant intensity I_0 into the integral expression for emissive power, and solve for E: E = I_0 \int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}}sin\theta\: d\theta\, d\phi Now, we will solve the integral to obtain the emissive power in terms of the constant intensity I_0:

Evaluate the Integral

E = I_0 \int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}}sin\theta\: d\theta\, d\phi = I_0 \int_{0}^{2\pi} d\phi \int_{0}^{\frac{\pi}{2}}sin\theta\: d\theta Separate the integral into two parts and solve them independently: E = I_0 \int_{0}^{2\pi} d\phi \int_{0}^{\frac{\pi}{2}}sin\theta\: d\theta = I_0 [2\pi] [-cos\theta |_0^{\frac{\pi}{2}}] E = I_0 [2\pi] (1-0) = 2\pi I_0 Thus, for a diffusely emitting surface, the emissive power (E) is related to the intensity of emitted radiation (I_0) by: E = 2\pi I_0

Key concepts

Radiation Intensity

When trying to understand how much energy is transmitted through radiation, it's essential to grasp the concept of radiation intensity. Often denoted as 'I', radiation intensity refers to the amount of energy radiated in the form of light or other types of electromagnetic waves per unit time, per unit surface area, per unit solid angle, and in a specified direction. This value can be mathematically expressed with the equation:

\[\begin{equation}I = \frac{dP}{dA \cdot d\Omega}\end{equation}\]
Where:

• dP represents the infinitesimal power emitted.
• dA symbolizes the infinitesimal area of the emitting surface that's perpendicular to the direction of radiation.
• d\Omega is the differential solid angle through which the power is being radiated.

In simpler terms, imagine you're holding a flashlight. The radiation intensity would describe how bright the beam is at any point in space, taking into consideration the spread of the light across a certain angle and area.

Emissive Power

Emissive power, sometimes referred to as radiant exitance or emitted flux density, is a crucial aspect of thermal radiation. It is defined as the total power emitted by a surface per unit area. For physicists and engineers dealing with heat transfer, emissive power (E) provides critical information about how a surface is performing enerically. We can calculate it using the integration of the radiation intensity over all directions above the surface within a hemisphere, represented by:

\[\begin{equation}E = \int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}} I(\theta, \phi)\, \sin\theta\: d\theta\, d\phi\end{equation}\]
Here, the variables \(\theta\) and \(\phi\) are the polar and azimuthal angles, respectively, defining the direction of emission. In practice, this means if you cover a light bulb with a frosted glass, the emissive power would be how much light power per area the glass is spreading into the room.

Diffuse Emission

In thermal radiation, a diffuse emitter is a surface that radiates energy uniformly in all directions. This kind of emission is a significant simplification used in many engineering and physics problems because it negates the need to account for directional dependence in radiation intensity. For a perfectly diffuse emitter, the intensity, denoted as I_0, remains constant no matter the angle of observation:

\[\begin{equation}I(\theta, \phi) = I_0\end{equation}\]
This can be thought of in reference to a perfectly white wall that reflects light the same way no matter where you stand in the room. The key takeaway here is that with diffuse emission, finding the overall emissive power becomes more straightforward because the constant nature of intensity allows for the integration over solid angles to be much less complex.

Solid Angle

As important as square meters are for measuring a wall's area, the concept of a solid angle is essential for understanding the dimensions of a 3D angle. A solid angle, represented by the symbol \Omega, is to three dimensions what an ordinary angle is to two dimensions. It's measured in steradians (sr), and one way to envision it is by picturing a cone projecting from a central point (like the tip of a sundae cone), with the solid angle being the 'space' it occupies. Mathematically, a solid angle can be defined as the area on the surface of a sphere that is covered by the cone, divided by the square of the sphere's radius:

\[\begin{equation}\Omega = \frac{A}{r^2}\end{equation}\]
where A is the spherical surface area, and r is the radius of the sphere. An entire sphere encompasses a solid angle of 4\pi steradians. Whether you're looking at a streetlight or the sun, measuring how much light or radiation covers a particular region of space requires understanding the solid angle it spans.